3.1100 \(\int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx\)

Optimal. Leaf size=321 \[ -\frac{(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}+\frac{10 b^9 B (b d-a e)}{e^{12} (d+e x)}-\frac{45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac{40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac{105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac{252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac{35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac{120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac{45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac{10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac{B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B \log (d+e x)}{e^{12}} \]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(11*e*(b*d - a*e)*(d + e*x)^11) - (B*(b*d - a*e)^10)/(10*e^12*(d + e*x)^10) + (10*
b*B*(b*d - a*e)^9)/(9*e^12*(d + e*x)^9) - (45*b^2*B*(b*d - a*e)^8)/(8*e^12*(d + e*x)^8) + (120*b^3*B*(b*d - a*
e)^7)/(7*e^12*(d + e*x)^7) - (35*b^4*B*(b*d - a*e)^6)/(e^12*(d + e*x)^6) + (252*b^5*B*(b*d - a*e)^5)/(5*e^12*(
d + e*x)^5) - (105*b^6*B*(b*d - a*e)^4)/(2*e^12*(d + e*x)^4) + (40*b^7*B*(b*d - a*e)^3)/(e^12*(d + e*x)^3) - (
45*b^8*B*(b*d - a*e)^2)/(2*e^12*(d + e*x)^2) + (10*b^9*B*(b*d - a*e))/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x])
/e^12

________________________________________________________________________________________

Rubi [A]  time = 0.576489, antiderivative size = 321, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 43} \[ -\frac{(a+b x)^{11} (B d-A e)}{11 e (d+e x)^{11} (b d-a e)}+\frac{10 b^9 B (b d-a e)}{e^{12} (d+e x)}-\frac{45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac{40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac{105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac{252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac{35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac{120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac{45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac{10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac{B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac{b^{10} B \log (d+e x)}{e^{12}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

-((B*d - A*e)*(a + b*x)^11)/(11*e*(b*d - a*e)*(d + e*x)^11) - (B*(b*d - a*e)^10)/(10*e^12*(d + e*x)^10) + (10*
b*B*(b*d - a*e)^9)/(9*e^12*(d + e*x)^9) - (45*b^2*B*(b*d - a*e)^8)/(8*e^12*(d + e*x)^8) + (120*b^3*B*(b*d - a*
e)^7)/(7*e^12*(d + e*x)^7) - (35*b^4*B*(b*d - a*e)^6)/(e^12*(d + e*x)^6) + (252*b^5*B*(b*d - a*e)^5)/(5*e^12*(
d + e*x)^5) - (105*b^6*B*(b*d - a*e)^4)/(2*e^12*(d + e*x)^4) + (40*b^7*B*(b*d - a*e)^3)/(e^12*(d + e*x)^3) - (
45*b^8*B*(b*d - a*e)^2)/(2*e^12*(d + e*x)^2) + (10*b^9*B*(b*d - a*e))/(e^12*(d + e*x)) + (b^10*B*Log[d + e*x])
/e^12

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(a+b x)^{10} (A+B x)}{(d+e x)^{12}} \, dx &=-\frac{(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}+\frac{B \int \frac{(a+b x)^{10}}{(d+e x)^{11}} \, dx}{e}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}+\frac{B \int \left (\frac{(-b d+a e)^{10}}{e^{10} (d+e x)^{11}}-\frac{10 b (b d-a e)^9}{e^{10} (d+e x)^{10}}+\frac{45 b^2 (b d-a e)^8}{e^{10} (d+e x)^9}-\frac{120 b^3 (b d-a e)^7}{e^{10} (d+e x)^8}+\frac{210 b^4 (b d-a e)^6}{e^{10} (d+e x)^7}-\frac{252 b^5 (b d-a e)^5}{e^{10} (d+e x)^6}+\frac{210 b^6 (b d-a e)^4}{e^{10} (d+e x)^5}-\frac{120 b^7 (b d-a e)^3}{e^{10} (d+e x)^4}+\frac{45 b^8 (b d-a e)^2}{e^{10} (d+e x)^3}-\frac{10 b^9 (b d-a e)}{e^{10} (d+e x)^2}+\frac{b^{10}}{e^{10} (d+e x)}\right ) \, dx}{e}\\ &=-\frac{(B d-A e) (a+b x)^{11}}{11 e (b d-a e) (d+e x)^{11}}-\frac{B (b d-a e)^{10}}{10 e^{12} (d+e x)^{10}}+\frac{10 b B (b d-a e)^9}{9 e^{12} (d+e x)^9}-\frac{45 b^2 B (b d-a e)^8}{8 e^{12} (d+e x)^8}+\frac{120 b^3 B (b d-a e)^7}{7 e^{12} (d+e x)^7}-\frac{35 b^4 B (b d-a e)^6}{e^{12} (d+e x)^6}+\frac{252 b^5 B (b d-a e)^5}{5 e^{12} (d+e x)^5}-\frac{105 b^6 B (b d-a e)^4}{2 e^{12} (d+e x)^4}+\frac{40 b^7 B (b d-a e)^3}{e^{12} (d+e x)^3}-\frac{45 b^8 B (b d-a e)^2}{2 e^{12} (d+e x)^2}+\frac{10 b^9 B (b d-a e)}{e^{12} (d+e x)}+\frac{b^{10} B \log (d+e x)}{e^{12}}\\ \end{align*}

Mathematica [B]  time = 1.91744, size = 1443, normalized size = 4.5 \[ -\frac{-27720 b^{10} B \log (d+e x) (d+e x)^{11}+252 a^{10} e^{10} (10 A e+B (d+11 e x))+280 a^9 b e^9 \left (9 A e (d+11 e x)+2 B \left (d^2+11 e x d+55 e^2 x^2\right )\right )+315 a^8 b^2 e^8 \left (8 A e \left (d^2+11 e x d+55 e^2 x^2\right )+3 B \left (d^3+11 e x d^2+55 e^2 x^2 d+165 e^3 x^3\right )\right )+360 a^7 b^3 e^7 \left (7 A e \left (d^3+11 e x d^2+55 e^2 x^2 d+165 e^3 x^3\right )+4 B \left (d^4+11 e x d^3+55 e^2 x^2 d^2+165 e^3 x^3 d+330 e^4 x^4\right )\right )+420 a^6 b^4 e^6 \left (6 A e \left (d^4+11 e x d^3+55 e^2 x^2 d^2+165 e^3 x^3 d+330 e^4 x^4\right )+5 B \left (d^5+11 e x d^4+55 e^2 x^2 d^3+165 e^3 x^3 d^2+330 e^4 x^4 d+462 e^5 x^5\right )\right )+504 a^5 b^5 e^5 \left (5 A e \left (d^5+11 e x d^4+55 e^2 x^2 d^3+165 e^3 x^3 d^2+330 e^4 x^4 d+462 e^5 x^5\right )+6 B \left (d^6+11 e x d^5+55 e^2 x^2 d^4+165 e^3 x^3 d^3+330 e^4 x^4 d^2+462 e^5 x^5 d+462 e^6 x^6\right )\right )+630 a^4 b^6 e^4 \left (4 A e \left (d^6+11 e x d^5+55 e^2 x^2 d^4+165 e^3 x^3 d^3+330 e^4 x^4 d^2+462 e^5 x^5 d+462 e^6 x^6\right )+7 B \left (d^7+11 e x d^6+55 e^2 x^2 d^5+165 e^3 x^3 d^4+330 e^4 x^4 d^3+462 e^5 x^5 d^2+462 e^6 x^6 d+330 e^7 x^7\right )\right )+840 a^3 b^7 e^3 \left (3 A e \left (d^7+11 e x d^6+55 e^2 x^2 d^5+165 e^3 x^3 d^4+330 e^4 x^4 d^3+462 e^5 x^5 d^2+462 e^6 x^6 d+330 e^7 x^7\right )+8 B \left (d^8+11 e x d^7+55 e^2 x^2 d^6+165 e^3 x^3 d^5+330 e^4 x^4 d^4+462 e^5 x^5 d^3+462 e^6 x^6 d^2+330 e^7 x^7 d+165 e^8 x^8\right )\right )+1260 a^2 b^8 e^2 \left (2 A e \left (d^8+11 e x d^7+55 e^2 x^2 d^6+165 e^3 x^3 d^5+330 e^4 x^4 d^4+462 e^5 x^5 d^3+462 e^6 x^6 d^2+330 e^7 x^7 d+165 e^8 x^8\right )+9 B \left (d^9+11 e x d^8+55 e^2 x^2 d^7+165 e^3 x^3 d^6+330 e^4 x^4 d^5+462 e^5 x^5 d^4+462 e^6 x^6 d^3+330 e^7 x^7 d^2+165 e^8 x^8 d+55 e^9 x^9\right )\right )+2520 a b^9 e \left (A e \left (d^9+11 e x d^8+55 e^2 x^2 d^7+165 e^3 x^3 d^6+330 e^4 x^4 d^5+462 e^5 x^5 d^4+462 e^6 x^6 d^3+330 e^7 x^7 d^2+165 e^8 x^8 d+55 e^9 x^9\right )+10 B \left (d^{10}+11 e x d^9+55 e^2 x^2 d^8+165 e^3 x^3 d^7+330 e^4 x^4 d^6+462 e^5 x^5 d^5+462 e^6 x^6 d^4+330 e^7 x^7 d^3+165 e^8 x^8 d^2+55 e^9 x^9 d+11 e^{10} x^{10}\right )\right )+b^{10} \left (2520 A e \left (d^{10}+11 e x d^9+55 e^2 x^2 d^8+165 e^3 x^3 d^7+330 e^4 x^4 d^6+462 e^5 x^5 d^5+462 e^6 x^6 d^4+330 e^7 x^7 d^3+165 e^8 x^8 d^2+55 e^9 x^9 d+11 e^{10} x^{10}\right )-B d \left (83711 d^{10}+893101 e x d^9+4313045 e^2 x^2 d^8+12430935 e^3 x^3 d^7+23718420 e^4 x^4 d^6+31376268 e^5 x^5 d^5+29241828 e^6 x^6 d^4+19057500 e^7 x^7 d^3+8385300 e^8 x^8 d^2+2286900 e^9 x^9 d+304920 e^{10} x^{10}\right )\right )}{27720 e^{12} (d+e x)^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^10*(A + B*x))/(d + e*x)^12,x]

[Out]

-(252*a^10*e^10*(10*A*e + B*(d + 11*e*x)) + 280*a^9*b*e^9*(9*A*e*(d + 11*e*x) + 2*B*(d^2 + 11*d*e*x + 55*e^2*x
^2)) + 315*a^8*b^2*e^8*(8*A*e*(d^2 + 11*d*e*x + 55*e^2*x^2) + 3*B*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x
^3)) + 360*a^7*b^3*e^7*(7*A*e*(d^3 + 11*d^2*e*x + 55*d*e^2*x^2 + 165*e^3*x^3) + 4*B*(d^4 + 11*d^3*e*x + 55*d^2
*e^2*x^2 + 165*d*e^3*x^3 + 330*e^4*x^4)) + 420*a^6*b^4*e^6*(6*A*e*(d^4 + 11*d^3*e*x + 55*d^2*e^2*x^2 + 165*d*e
^3*x^3 + 330*e^4*x^4) + 5*B*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5
)) + 504*a^5*b^5*e^5*(5*A*e*(d^5 + 11*d^4*e*x + 55*d^3*e^2*x^2 + 165*d^2*e^3*x^3 + 330*d*e^4*x^4 + 462*e^5*x^5
) + 6*B*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x^5 + 462*e^6*x^6))
+ 630*a^4*b^6*e^4*(4*A*e*(d^6 + 11*d^5*e*x + 55*d^4*e^2*x^2 + 165*d^3*e^3*x^3 + 330*d^2*e^4*x^4 + 462*d*e^5*x
^5 + 462*e^6*x^6) + 7*B*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x
^5 + 462*d*e^6*x^6 + 330*e^7*x^7)) + 840*a^3*b^7*e^3*(3*A*e*(d^7 + 11*d^6*e*x + 55*d^5*e^2*x^2 + 165*d^4*e^3*x
^3 + 330*d^3*e^4*x^4 + 462*d^2*e^5*x^5 + 462*d*e^6*x^6 + 330*e^7*x^7) + 8*B*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2
+ 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5 + 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8)) + 126
0*a^2*b^8*e^2*(2*A*e*(d^8 + 11*d^7*e*x + 55*d^6*e^2*x^2 + 165*d^5*e^3*x^3 + 330*d^4*e^4*x^4 + 462*d^3*e^5*x^5
+ 462*d^2*e^6*x^6 + 330*d*e^7*x^7 + 165*e^8*x^8) + 9*B*(d^9 + 11*d^8*e*x + 55*d^7*e^2*x^2 + 165*d^6*e^3*x^3 +
330*d^5*e^4*x^4 + 462*d^4*e^5*x^5 + 462*d^3*e^6*x^6 + 330*d^2*e^7*x^7 + 165*d*e^8*x^8 + 55*e^9*x^9)) + 2520*a*
b^9*e*(A*e*(d^9 + 11*d^8*e*x + 55*d^7*e^2*x^2 + 165*d^6*e^3*x^3 + 330*d^5*e^4*x^4 + 462*d^4*e^5*x^5 + 462*d^3*
e^6*x^6 + 330*d^2*e^7*x^7 + 165*d*e^8*x^8 + 55*e^9*x^9) + 10*B*(d^10 + 11*d^9*e*x + 55*d^8*e^2*x^2 + 165*d^7*e
^3*x^3 + 330*d^6*e^4*x^4 + 462*d^5*e^5*x^5 + 462*d^4*e^6*x^6 + 330*d^3*e^7*x^7 + 165*d^2*e^8*x^8 + 55*d*e^9*x^
9 + 11*e^10*x^10)) + b^10*(2520*A*e*(d^10 + 11*d^9*e*x + 55*d^8*e^2*x^2 + 165*d^7*e^3*x^3 + 330*d^6*e^4*x^4 +
462*d^5*e^5*x^5 + 462*d^4*e^6*x^6 + 330*d^3*e^7*x^7 + 165*d^2*e^8*x^8 + 55*d*e^9*x^9 + 11*e^10*x^10) - B*d*(83
711*d^10 + 893101*d^9*e*x + 4313045*d^8*e^2*x^2 + 12430935*d^7*e^3*x^3 + 23718420*d^6*e^4*x^4 + 31376268*d^5*e
^5*x^5 + 29241828*d^4*e^6*x^6 + 19057500*d^3*e^7*x^7 + 8385300*d^2*e^8*x^8 + 2286900*d*e^9*x^9 + 304920*e^10*x
^10)) - 27720*b^10*B*(d + e*x)^11*Log[d + e*x])/(27720*e^12*(d + e*x)^11)

________________________________________________________________________________________

Maple [B]  time = 0.015, size = 2907, normalized size = 9.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^10*(B*x+A)/(e*x+d)^12,x)

[Out]

-15*b^3/e^4/(e*x+d)^8*A*a^7+15*b^10/e^11/(e*x+d)^8*A*d^7-45/8*b^2/e^4/(e*x+d)^8*B*a^8-165/8*b^10/e^12/(e*x+d)^
8*B*d^8-1/e^2/(e*x+d)^10*A*a^9*b+1/e^11/(e*x+d)^10*A*d^9*b^10-11/10/e^12/(e*x+d)^10*b^10*B*d^10-30*b^7/e^8/(e*
x+d)^4*A*a^3+30*b^10/e^11/(e*x+d)^4*A*d^3-105/2*b^6/e^8/(e*x+d)^4*B*a^4-165/2*b^10/e^12/(e*x+d)^4*B*d^4-1/11/e
^11/(e*x+d)^11*A*d^10*b^10+1/11/e^2/(e*x+d)^11*B*d*a^10+1/11/e^12/(e*x+d)^11*b^10*B*d^11-42*b^6/e^7/(e*x+d)^5*
A*a^4-42*b^10/e^11/(e*x+d)^5*A*d^4-252/5*b^5/e^7/(e*x+d)^5*B*a^5+462/5*b^10/e^12/(e*x+d)^5*B*d^5-15*b^8/e^9/(e
*x+d)^3*A*a^2-15*b^10/e^11/(e*x+d)^3*A*d^2-1/10/e^2/(e*x+d)^10*B*a^10-1/11/e/(e*x+d)^11*a^10*A-b^10/e^11/(e*x+
d)*A+756*b^8/e^10/(e*x+d)^5*B*a^2*d^3-420*b^9/e^11/(e*x+d)^5*B*a*d^4+105*b^4/e^5/(e*x+d)^8*A*a^6*d-315*b^5/e^6
/(e*x+d)^8*A*a^5*d^2+525*b^6/e^7/(e*x+d)^8*A*a^4*d^3-525*b^7/e^8/(e*x+d)^8*A*a^3*d^4+315*b^8/e^9/(e*x+d)^8*A*a
^2*d^5-105*b^9/e^10/(e*x+d)^8*A*a*d^6-210*b^9/e^10/(e*x+d)^6*A*a*d^4+252*b^5/e^7/(e*x+d)^6*B*a^5*d-735*b^6/e^8
/(e*x+d)^6*B*a^4*d^2+1120*b^7/e^9/(e*x+d)^6*B*a^3*d^3-945*b^8/e^10/(e*x+d)^6*B*a^2*d^4+420*b^9/e^11/(e*x+d)^6*
B*a*d^5+30*b^9/e^10/(e*x+d)^3*A*a*d+135*b^8/e^10/(e*x+d)^3*B*a^2*d-150*b^9/e^11/(e*x+d)^3*B*a*d^2+180*b^5/e^6/
(e*x+d)^7*A*a^5*d+45/11/e^4/(e*x+d)^11*B*d^3*a^8*b^2-120/11/e^5/(e*x+d)^11*B*d^4*a^7*b^3+210/11/e^6/(e*x+d)^11
*B*d^5*a^6*b^4-252/11/e^7/(e*x+d)^11*B*d^6*a^5*b^5+210/11/e^8/(e*x+d)^11*B*d^7*a^4*b^6-120/11/e^9/(e*x+d)^11*B
*d^8*a^3*b^7+45/11/e^10/(e*x+d)^11*B*d^9*a^2*b^8-10/11/e^11/(e*x+d)^11*B*d^10*a*b^9+210*b^6/e^7/(e*x+d)^6*A*a^
4*d-420*b^7/e^8/(e*x+d)^6*A*a^3*d^2+420*b^8/e^9/(e*x+d)^6*A*a^2*d^3+120/11/e^4/(e*x+d)^11*A*d^3*a^7*b^3-210/11
/e^5/(e*x+d)^11*A*d^4*a^6*b^4+252/11/e^6/(e*x+d)^11*A*d^5*a^5*b^5-210/11/e^7/(e*x+d)^11*A*d^6*a^4*b^6+120/11/e
^8/(e*x+d)^11*A*d^7*a^3*b^7-45/11/e^9/(e*x+d)^11*A*d^8*a^2*b^8+10/11/e^10/(e*x+d)^11*A*d^9*a*b^9-10/11/e^3/(e*
x+d)^11*B*d^2*a^9*b-450*b^6/e^7/(e*x+d)^7*A*a^4*d^2+600*b^7/e^8/(e*x+d)^7*A*a^3*d^3-450*b^8/e^9/(e*x+d)^7*A*a^
2*d^4+180*b^9/e^10/(e*x+d)^7*A*a*d^5+150*b^4/e^6/(e*x+d)^7*B*a^6*d-540*b^5/e^7/(e*x+d)^7*B*a^5*d^2+1050*b^6/e^
8/(e*x+d)^7*B*a^4*d^3-1200*b^7/e^9/(e*x+d)^7*B*a^3*d^4+810*b^8/e^10/(e*x+d)^7*B*a^2*d^5-300*b^9/e^11/(e*x+d)^7
*B*a*d^6+40*b^3/e^4/(e*x+d)^9*A*a^7*d-140*b^4/e^5/(e*x+d)^9*A*a^6*d^2+280*b^5/e^6/(e*x+d)^9*A*a^5*d^3-350*b^6/
e^7/(e*x+d)^9*A*a^4*d^4+280*b^7/e^8/(e*x+d)^9*A*a^3*d^5-140*b^8/e^9/(e*x+d)^9*A*a^2*d^6+40*b^9/e^10/(e*x+d)^9*
A*a*d^7+15*b^2/e^4/(e*x+d)^9*B*a^8*d-80*b^3/e^5/(e*x+d)^9*B*a^7*d^2+700/3*b^4/e^6/(e*x+d)^9*B*a^6*d^3-420*b^5/
e^7/(e*x+d)^9*B*a^5*d^4+490*b^6/e^8/(e*x+d)^9*B*a^4*d^5-1120/3*b^7/e^9/(e*x+d)^9*B*a^3*d^6+180*b^8/e^10/(e*x+d
)^9*B*a^2*d^7-50*b^9/e^11/(e*x+d)^9*B*a*d^8+50*b^9/e^11/(e*x+d)^2*B*a*d+168*b^7/e^8/(e*x+d)^5*A*a^3*d-252*b^8/
e^9/(e*x+d)^5*A*a^2*d^2+168*b^9/e^10/(e*x+d)^5*A*a*d^3+294*b^6/e^8/(e*x+d)^5*B*a^4*d-672*b^7/e^9/(e*x+d)^5*B*a
^3*d^2-40*b^7/e^9/(e*x+d)^3*B*a^3+55*b^10/e^12/(e*x+d)^3*B*d^3+330/7*b^10/e^12/(e*x+d)^7*B*d^7-5*b^2/e^3/(e*x+
d)^9*A*a^8-5*b^10/e^11/(e*x+d)^9*A*d^8-10/9*b/e^3/(e*x+d)^9*B*a^9+55/9*b^10/e^12/(e*x+d)^9*B*d^9-5*b^9/e^10/(e
*x+d)^2*A*a+5*b^10/e^11/(e*x+d)^2*A*d-45/2*b^8/e^10/(e*x+d)^2*B*a^2-55/2*b^10/e^12/(e*x+d)^2*B*d^2-10*b^9/e^11
/(e*x+d)*B*a+11*b^10/e^12/(e*x+d)*B*d-30*b^4/e^5/(e*x+d)^7*A*a^6-30*b^10/e^11/(e*x+d)^7*A*d^6-120/7*b^3/e^5/(e
*x+d)^7*B*a^7-42*b^5/e^6/(e*x+d)^6*A*a^5+42*b^10/e^11/(e*x+d)^6*A*d^5-35*b^4/e^6/(e*x+d)^6*B*a^6-77*b^10/e^12/
(e*x+d)^6*B*d^6+60*b^3/e^5/(e*x+d)^8*B*a^7*d-525/2*b^4/e^6/(e*x+d)^8*B*a^6*d^2+630*b^5/e^7/(e*x+d)^8*B*a^5*d^3
-3675/4*b^6/e^8/(e*x+d)^8*B*a^4*d^4+840*b^7/e^9/(e*x+d)^8*B*a^3*d^5-945/2*b^8/e^10/(e*x+d)^8*B*a^2*d^6+150*b^9
/e^11/(e*x+d)^8*B*a*d^7+9/e^3/(e*x+d)^10*A*d*a^8*b^2-36/e^4/(e*x+d)^10*A*d^2*a^7*b^3+84/e^5/(e*x+d)^10*A*d^3*a
^6*b^4-126/e^6/(e*x+d)^10*A*d^4*a^5*b^5+126/e^7/(e*x+d)^10*A*d^5*a^4*b^6-84/e^8/(e*x+d)^10*A*d^6*a^3*b^7+36/e^
9/(e*x+d)^10*A*d^7*a^2*b^8-9/e^10/(e*x+d)^10*A*d^8*a*b^9+2/e^3/(e*x+d)^10*B*d*a^9*b-27/2/e^4/(e*x+d)^10*B*d^2*
a^8*b^2+48/e^5/(e*x+d)^10*B*d^3*a^7*b^3-105/e^6/(e*x+d)^10*B*d^4*a^6*b^4+756/5/e^7/(e*x+d)^10*B*d^5*a^5*b^5-14
7/e^8/(e*x+d)^10*B*d^6*a^4*b^6+96/e^9/(e*x+d)^10*B*d^7*a^3*b^7-81/2/e^10/(e*x+d)^10*B*d^8*a^2*b^8+10/e^11/(e*x
+d)^10*B*d^9*a*b^9+90*b^8/e^9/(e*x+d)^4*A*a^2*d-90*b^9/e^10/(e*x+d)^4*A*a*d^2+240*b^7/e^9/(e*x+d)^4*B*a^3*d-40
5*b^8/e^10/(e*x+d)^4*B*a^2*d^2+300*b^9/e^11/(e*x+d)^4*B*a*d^3+10/11/e^2/(e*x+d)^11*A*d*a^9*b-45/11/e^3/(e*x+d)
^11*A*d^2*a^8*b^2+b^10*B*ln(e*x+d)/e^12

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Maxima [B]  time = 4.07874, size = 2608, normalized size = 8.12 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="maxima")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e
^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*
b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(10*B*a*
b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^1
0)*d^2*e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*
d^4*e^7 - 12*(10*B*a*b^9 + A*b^10)*d^3*e^8 - 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^
4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d*e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a*b^9 + A*b
^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1
980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 - 70*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^1
1)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^4*e^7 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 315*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
- 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 420*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^4*e^7 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*
B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x)/(e^23*x^11 + 11*d*e^22*x^10 + 55*d^2*e^21*x
^9 + 165*d^3*e^20*x^8 + 330*d^4*e^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*e^17*x^5 + 330*d^7*e^16*x^4 + 165*d^8*e^
15*x^3 + 55*d^9*e^14*x^2 + 11*d^10*e^13*x + d^11*e^12) + B*b^10*log(e*x + d)/e^12

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Fricas [B]  time = 2.15747, size = 4635, normalized size = 14.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="fricas")

[Out]

1/27720*(83711*B*b^10*d^11 - 2520*A*a^10*e^11 - 2520*(10*B*a*b^9 + A*b^10)*d^10*e - 1260*(9*B*a^2*b^8 + 2*A*a*
b^9)*d^9*e^2 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^8*e^3 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^7*e^4 - 504*(6*B*a^
5*b^5 + 5*A*a^4*b^6)*d^6*e^5 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^5*e^6 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^4*e
^7 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^3*e^8 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*d^2*e^9 - 252*(B*a^10 + 10*A*a^9*
b)*d*e^10 + 27720*(11*B*b^10*d*e^10 - (10*B*a*b^9 + A*b^10)*e^11)*x^10 + 69300*(33*B*b^10*d^2*e^9 - 2*(10*B*a*
b^9 + A*b^10)*d*e^10 - (9*B*a^2*b^8 + 2*A*a*b^9)*e^11)*x^9 + 69300*(121*B*b^10*d^3*e^8 - 6*(10*B*a*b^9 + A*b^1
0)*d^2*e^9 - 3*(9*B*a^2*b^8 + 2*A*a*b^9)*d*e^10 - 2*(8*B*a^3*b^7 + 3*A*a^2*b^8)*e^11)*x^8 + 69300*(275*B*b^10*
d^4*e^7 - 12*(10*B*a*b^9 + A*b^10)*d^3*e^8 - 6*(9*B*a^2*b^8 + 2*A*a*b^9)*d^2*e^9 - 4*(8*B*a^3*b^7 + 3*A*a^2*b^
8)*d*e^10 - 3*(7*B*a^4*b^6 + 4*A*a^3*b^7)*e^11)*x^7 + 19404*(1507*B*b^10*d^5*e^6 - 60*(10*B*a*b^9 + A*b^10)*d^
4*e^7 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^3*e^8 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^2*e^9 - 15*(7*B*a^4*b^6 + 4*A*
a^3*b^7)*d*e^10 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*e^11)*x^6 + 19404*(1617*B*b^10*d^6*e^5 - 60*(10*B*a*b^9 + A*b
^10)*d^5*e^6 - 30*(9*B*a^2*b^8 + 2*A*a*b^9)*d^4*e^7 - 20*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^3*e^8 - 15*(7*B*a^4*b^6
+ 4*A*a^3*b^7)*d^2*e^9 - 12*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d*e^10 - 10*(5*B*a^6*b^4 + 6*A*a^5*b^5)*e^11)*x^5 + 1
980*(11979*B*b^10*d^7*e^4 - 420*(10*B*a*b^9 + A*b^10)*d^6*e^5 - 210*(9*B*a^2*b^8 + 2*A*a*b^9)*d^5*e^6 - 140*(8
*B*a^3*b^7 + 3*A*a^2*b^8)*d^4*e^7 - 105*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^3*e^8 - 84*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d
^2*e^9 - 70*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d*e^10 - 60*(4*B*a^7*b^3 + 7*A*a^6*b^4)*e^11)*x^4 + 495*(25113*B*b^10*
d^8*e^3 - 840*(10*B*a*b^9 + A*b^10)*d^7*e^4 - 420*(9*B*a^2*b^8 + 2*A*a*b^9)*d^6*e^5 - 280*(8*B*a^3*b^7 + 3*A*a
^2*b^8)*d^5*e^6 - 210*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^4*e^7 - 168*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^3*e^8 - 140*(5*B
*a^6*b^4 + 6*A*a^5*b^5)*d^2*e^9 - 120*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d*e^10 - 105*(3*B*a^8*b^2 + 8*A*a^7*b^3)*e^1
1)*x^3 + 55*(78419*B*b^10*d^9*e^2 - 2520*(10*B*a*b^9 + A*b^10)*d^8*e^3 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^7*e^
4 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^6*e^5 - 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^5*e^6 - 504*(6*B*a^5*b^5 + 5*A
*a^4*b^6)*d^4*e^7 - 420*(5*B*a^6*b^4 + 6*A*a^5*b^5)*d^3*e^8 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^2*e^9 - 315*(3
*B*a^8*b^2 + 8*A*a^7*b^3)*d*e^10 - 280*(2*B*a^9*b + 9*A*a^8*b^2)*e^11)*x^2 + 11*(81191*B*b^10*d^10*e - 2520*(1
0*B*a*b^9 + A*b^10)*d^9*e^2 - 1260*(9*B*a^2*b^8 + 2*A*a*b^9)*d^8*e^3 - 840*(8*B*a^3*b^7 + 3*A*a^2*b^8)*d^7*e^4
- 630*(7*B*a^4*b^6 + 4*A*a^3*b^7)*d^6*e^5 - 504*(6*B*a^5*b^5 + 5*A*a^4*b^6)*d^5*e^6 - 420*(5*B*a^6*b^4 + 6*A*
a^5*b^5)*d^4*e^7 - 360*(4*B*a^7*b^3 + 7*A*a^6*b^4)*d^3*e^8 - 315*(3*B*a^8*b^2 + 8*A*a^7*b^3)*d^2*e^9 - 280*(2*
B*a^9*b + 9*A*a^8*b^2)*d*e^10 - 252*(B*a^10 + 10*A*a^9*b)*e^11)*x + 27720*(B*b^10*e^11*x^11 + 11*B*b^10*d*e^10
*x^10 + 55*B*b^10*d^2*e^9*x^9 + 165*B*b^10*d^3*e^8*x^8 + 330*B*b^10*d^4*e^7*x^7 + 462*B*b^10*d^5*e^6*x^6 + 462
*B*b^10*d^6*e^5*x^5 + 330*B*b^10*d^7*e^4*x^4 + 165*B*b^10*d^8*e^3*x^3 + 55*B*b^10*d^9*e^2*x^2 + 11*B*b^10*d^10
*e*x + B*b^10*d^11)*log(e*x + d))/(e^23*x^11 + 11*d*e^22*x^10 + 55*d^2*e^21*x^9 + 165*d^3*e^20*x^8 + 330*d^4*e
^19*x^7 + 462*d^5*e^18*x^6 + 462*d^6*e^17*x^5 + 330*d^7*e^16*x^4 + 165*d^8*e^15*x^3 + 55*d^9*e^14*x^2 + 11*d^1
0*e^13*x + d^11*e^12)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**10*(B*x+A)/(e*x+d)**12,x)

[Out]

Timed out

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Giac [B]  time = 1.621, size = 2488, normalized size = 7.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^10*(B*x+A)/(e*x+d)^12,x, algorithm="giac")

[Out]

B*b^10*e^(-12)*log(abs(x*e + d)) + 1/27720*(27720*(11*B*b^10*d*e^9 - 10*B*a*b^9*e^10 - A*b^10*e^10)*x^10 + 693
00*(33*B*b^10*d^2*e^8 - 20*B*a*b^9*d*e^9 - 2*A*b^10*d*e^9 - 9*B*a^2*b^8*e^10 - 2*A*a*b^9*e^10)*x^9 + 69300*(12
1*B*b^10*d^3*e^7 - 60*B*a*b^9*d^2*e^8 - 6*A*b^10*d^2*e^8 - 27*B*a^2*b^8*d*e^9 - 6*A*a*b^9*d*e^9 - 16*B*a^3*b^7
*e^10 - 6*A*a^2*b^8*e^10)*x^8 + 69300*(275*B*b^10*d^4*e^6 - 120*B*a*b^9*d^3*e^7 - 12*A*b^10*d^3*e^7 - 54*B*a^2
*b^8*d^2*e^8 - 12*A*a*b^9*d^2*e^8 - 32*B*a^3*b^7*d*e^9 - 12*A*a^2*b^8*d*e^9 - 21*B*a^4*b^6*e^10 - 12*A*a^3*b^7
*e^10)*x^7 + 19404*(1507*B*b^10*d^5*e^5 - 600*B*a*b^9*d^4*e^6 - 60*A*b^10*d^4*e^6 - 270*B*a^2*b^8*d^3*e^7 - 60
*A*a*b^9*d^3*e^7 - 160*B*a^3*b^7*d^2*e^8 - 60*A*a^2*b^8*d^2*e^8 - 105*B*a^4*b^6*d*e^9 - 60*A*a^3*b^7*d*e^9 - 7
2*B*a^5*b^5*e^10 - 60*A*a^4*b^6*e^10)*x^6 + 19404*(1617*B*b^10*d^6*e^4 - 600*B*a*b^9*d^5*e^5 - 60*A*b^10*d^5*e
^5 - 270*B*a^2*b^8*d^4*e^6 - 60*A*a*b^9*d^4*e^6 - 160*B*a^3*b^7*d^3*e^7 - 60*A*a^2*b^8*d^3*e^7 - 105*B*a^4*b^6
*d^2*e^8 - 60*A*a^3*b^7*d^2*e^8 - 72*B*a^5*b^5*d*e^9 - 60*A*a^4*b^6*d*e^9 - 50*B*a^6*b^4*e^10 - 60*A*a^5*b^5*e
^10)*x^5 + 1980*(11979*B*b^10*d^7*e^3 - 4200*B*a*b^9*d^6*e^4 - 420*A*b^10*d^6*e^4 - 1890*B*a^2*b^8*d^5*e^5 - 4
20*A*a*b^9*d^5*e^5 - 1120*B*a^3*b^7*d^4*e^6 - 420*A*a^2*b^8*d^4*e^6 - 735*B*a^4*b^6*d^3*e^7 - 420*A*a^3*b^7*d^
3*e^7 - 504*B*a^5*b^5*d^2*e^8 - 420*A*a^4*b^6*d^2*e^8 - 350*B*a^6*b^4*d*e^9 - 420*A*a^5*b^5*d*e^9 - 240*B*a^7*
b^3*e^10 - 420*A*a^6*b^4*e^10)*x^4 + 495*(25113*B*b^10*d^8*e^2 - 8400*B*a*b^9*d^7*e^3 - 840*A*b^10*d^7*e^3 - 3
780*B*a^2*b^8*d^6*e^4 - 840*A*a*b^9*d^6*e^4 - 2240*B*a^3*b^7*d^5*e^5 - 840*A*a^2*b^8*d^5*e^5 - 1470*B*a^4*b^6*
d^4*e^6 - 840*A*a^3*b^7*d^4*e^6 - 1008*B*a^5*b^5*d^3*e^7 - 840*A*a^4*b^6*d^3*e^7 - 700*B*a^6*b^4*d^2*e^8 - 840
*A*a^5*b^5*d^2*e^8 - 480*B*a^7*b^3*d*e^9 - 840*A*a^6*b^4*d*e^9 - 315*B*a^8*b^2*e^10 - 840*A*a^7*b^3*e^10)*x^3
+ 55*(78419*B*b^10*d^9*e - 25200*B*a*b^9*d^8*e^2 - 2520*A*b^10*d^8*e^2 - 11340*B*a^2*b^8*d^7*e^3 - 2520*A*a*b^
9*d^7*e^3 - 6720*B*a^3*b^7*d^6*e^4 - 2520*A*a^2*b^8*d^6*e^4 - 4410*B*a^4*b^6*d^5*e^5 - 2520*A*a^3*b^7*d^5*e^5
- 3024*B*a^5*b^5*d^4*e^6 - 2520*A*a^4*b^6*d^4*e^6 - 2100*B*a^6*b^4*d^3*e^7 - 2520*A*a^5*b^5*d^3*e^7 - 1440*B*a
^7*b^3*d^2*e^8 - 2520*A*a^6*b^4*d^2*e^8 - 945*B*a^8*b^2*d*e^9 - 2520*A*a^7*b^3*d*e^9 - 560*B*a^9*b*e^10 - 2520
*A*a^8*b^2*e^10)*x^2 + 11*(81191*B*b^10*d^10 - 25200*B*a*b^9*d^9*e - 2520*A*b^10*d^9*e - 11340*B*a^2*b^8*d^8*e
^2 - 2520*A*a*b^9*d^8*e^2 - 6720*B*a^3*b^7*d^7*e^3 - 2520*A*a^2*b^8*d^7*e^3 - 4410*B*a^4*b^6*d^6*e^4 - 2520*A*
a^3*b^7*d^6*e^4 - 3024*B*a^5*b^5*d^5*e^5 - 2520*A*a^4*b^6*d^5*e^5 - 2100*B*a^6*b^4*d^4*e^6 - 2520*A*a^5*b^5*d^
4*e^6 - 1440*B*a^7*b^3*d^3*e^7 - 2520*A*a^6*b^4*d^3*e^7 - 945*B*a^8*b^2*d^2*e^8 - 2520*A*a^7*b^3*d^2*e^8 - 560
*B*a^9*b*d*e^9 - 2520*A*a^8*b^2*d*e^9 - 252*B*a^10*e^10 - 2520*A*a^9*b*e^10)*x + (83711*B*b^10*d^11 - 25200*B*
a*b^9*d^10*e - 2520*A*b^10*d^10*e - 11340*B*a^2*b^8*d^9*e^2 - 2520*A*a*b^9*d^9*e^2 - 6720*B*a^3*b^7*d^8*e^3 -
2520*A*a^2*b^8*d^8*e^3 - 4410*B*a^4*b^6*d^7*e^4 - 2520*A*a^3*b^7*d^7*e^4 - 3024*B*a^5*b^5*d^6*e^5 - 2520*A*a^4
*b^6*d^6*e^5 - 2100*B*a^6*b^4*d^5*e^6 - 2520*A*a^5*b^5*d^5*e^6 - 1440*B*a^7*b^3*d^4*e^7 - 2520*A*a^6*b^4*d^4*e
^7 - 945*B*a^8*b^2*d^3*e^8 - 2520*A*a^7*b^3*d^3*e^8 - 560*B*a^9*b*d^2*e^9 - 2520*A*a^8*b^2*d^2*e^9 - 252*B*a^1
0*d*e^10 - 2520*A*a^9*b*d*e^10 - 2520*A*a^10*e^11)*e^(-1))*e^(-11)/(x*e + d)^11